Factorization of quadratic equation is another very important process in algebra which means breaking a quadratic equation into its linear factors. This technique is useful for solving quadratic equations, rewriting and simplifying algebraic expressions, or visualizing quadratic functions and their properties.

A quadratic equation is of the form of ax² + bx + c= 0 with a, b, and c being real numbers or constants.

One way to perform it is by factorizing these equations so that it becomes easier to solve for them, and for more understanding of their roots and behaviors as well.

What is Factorization of Quadratic Equations?

In factorization of quadratic equations, it is the process of putting a quadratic expression in the form of a product of two binomials at most. This means transforming an equation such as ax² + bx + c = 0 to a form K (px + q)(rx + s) = 0. This process is important because after completing this process we have to find the value of x that makes the coefficient of x in each binomial to be equal to zero to solve the quadratic equation.

Factorization of Quadratic Equations

These are the coefficients of the quadratic equation as the roots of the equation correspond to certain values. This process is not only used to solve quadratic equations, but it is also crucial when it comes to understanding the geometric interpretation of quadratic functions like the x-intercepts of quadratic functions represented on a graph. Quadratic equations factorization may involve different strategies and techniques depending on the specific situation or problem.

Factorization of Quadratic Equations

There are several methods to factorize quadratic equations, each suited for different types of equations. The primary methods include:

• Factorization by Splitting the Middle Term
• Factoring Using Formula
• Factoring Using the Quadratic Formula
• Factoring Using Algebraic Identities

Steps for Factorization of Quadratic Equation by Splitting Middle Term

This one involves splitting the middle term with x, so that when you have added the coefficients of the split terms they should be equal to the middle coefficient and the product of these two terms should equal the product of the first and the last coefficients.

Step 1: Identify the quadratic equation in the form ax² + bx + c

Step 2: Find two numbers whose sum is b (the coefficient of x) and whose product is ac (the product of the coefficients of x² and the constant term).

Step 3: Rewrite the middle term using these two numbers.

Step 4: Group the terms into two pairs.

Step 5: Factor out the common factor from each pair.

Step 6: Factor out the common binomial factor from the resulting expression.

Example: Factorize 2x² + 7x + 3.

Identify the quadratic equation: 2x² + 7x + 3

Find two numbers: We need two numbers that add to 7 and multiply by 2 * 3 = 6. These numbers are 6 and 1

Rewrite the middle term: 2x² + 6x + x + 3

Group the terms: (2x² + 6x) + (x + 3)

Factor out the common factor: 2x(x + 3) + 1(x + 3)

Factor out the common binomial factor: (2x + 1)(x + 3)

Thus, 2x² + 7x + 3 = (2x + 1)(x + 3)

Steps for Factoring Quadratic Equation using Formula

This method uses a specific formula to factorize the quadratic equation.

Step 1: Identify the quadratic equation in the form ax² + bx + c.

Step 2: Use the factorization formula: If a ≠ 1, use the formula for ax² + bx + c.

Step 3: Rewrite the equation based on the identified formula.

Step 4: Simplify to find the factors.

Example: Factorize x² + 5x + 6.

Identify the quadratic equation: x² + 5x + 6

Use the factorization formula: x² + (a+b)x + ab = (x+a)(x+b)

Here, a = 2 and b = 3, since 2+3 = 5 and 2×3 = 6

Rewrite the equation: (x+2)(x+3)

Thus, x² + 5x + 6 = (x+2)(x+3).

Steps for Factoring Quadratic Equation using Quadratic Formula

Quadratic formula is a standard method for solving any quadratic equation and can also be used for factorization. The quadratic formula is derived from the process of completing the square and is given by:

x = (-b ± √(b² – 4ac)) / (2a)

This formula gives the roots of the quadratic equation, which can then be used to factorize the quadratic expression.

Step 1: Identify the quadratic equation in the form ax² + bx + c

Step 2: Apply the quadratic formula to find the roots x₁ and x₂

Step 3: Rewrite the quadratic equation as a(x – x₁)(x – x₂)

Example: Factorize 3x² – 2x – 8

Identify the quadratic equation:

3x² – 2x – 8

Apply the quadratic formula:

x = (-(-2) ± √((-2)² – 4.3.(-8)) / (2.3)

Simplify the discriminant

x = x = (2 ± √(4 + 96)) / 6 = x = (2 ± √(100)) / 6 = x = (2 ± 10) / 6

Find the roots: x₁ = 2 and x₂ = – 4/3

Rewrite the equation: 3 (x – 2) (x + 4/3)

Thus, 3x² – 2x – 8 = 3 (x – 2) (x + 4/3)

Steps for Factoring Quadratic Equations using Algebraic Identities

Algebraic identities can simplify the factorization process, particularly when dealing with specific forms of quadratic equations. They provide a straightforward method to factorize certain types of quadratic expressions. Common identities include:

Step 1: Identify the quadratic equation in a recognizable identity form.

Step 2: Apply the appropriate identity to factorize the expression.

Example: Factorize 4x² – 9.

Identify the quadratic equation: 4x² – 9 is in the form a² – b²

Apply the identity: a² – b² = (a + b)(a – b), where a = 2x and b = 3

Rewrite the equation: (2x + 3)(2x – 3)

Thus, 4x² – 9 = (2x + 3)(2x – 3)

Application of Factorization of Quadratic Equations

The applications of Factorization of Quadratic equations are as follows:

• Solving Quadratic Equations: Factorization makes the process of finding the roots of quadratic equations easier, which is very important in algebra and calculus.

• Graphing Quadratic Functions: Knowledge of the factors assists when searching for the x-intercepts of the function which also assists in graph sketching.

• Physics Problems: Most of the projectiles are governed by quadratic equations and hence the function can model many physical occurrences.

• Optimization Problems: Quadratics have applications in finding the maximum and minimum values which are commonly used cases in the field of economics and engineering.

• Geometry: Factorization helps in solving geometric problems involving areas and dimensions.

Conclusion

Factorization of quadratic equations is an essential algebraic technique that simplifies solving and understanding quadratic equations. By breaking down a quadratic equation into simpler binomial factors, we can easily find the roots and analyze the properties of quadratic functions. It is therefore realized there are various techniques used in factorization.

Read More,

• Quadratic Function
• Quadratic Inequalities
• Solving Cubic Equations

Examples on Factorization of Quadratic Equations

Example 1: Factorize: x² – 5x + 6

x² – 5x + 6

⇒ x² – 2x – 3x + 6

⇒ x(x – 2) – 3(x – 2)

⇒ (x – 2)(x – 3)

Example 2: Factorize: x² – 6x + 9

x² – 6x + 9

⇒ x² – 2.3x + (3)²

Comparing with a² – 2ab + b² = (a – b)²

⇒ (x – 3)²

Example 3: Factorize: x² + x – 12

x² + x – 12

⇒ x² +4x – 3x + 12

⇒ x(x + 4) – 3(x + 4)

⇒ (x + 4)(x – 3)

Example 4: Factorize: x² + 8x + 16

x² + 8x + 16

⇒ x² + 2.4x + (4)²

Comparing with a² + 2ab + b² = (a + b)²

⇒ (x + 4)²

Factorization of Quadratic Equations – FAQs

When can you not Factorize a Quadratic?

You cannot factorize a quadratic if it doesn’t have rational roots: this happens when the discriminant ( b² – 4ac) is not a perfect square.

What are the Disadvantages of Factoring to Solve Quadratic Equations?

A disadvantage of factoring is that it is very time-consuming and slightly intricate especially when large coefficients are involved or if the roots are irrational.

What is the Best way to Factor a Quadratic Equation?

Best way is to find two numbers that multiply to the constant term (ac) and add up to the linear coefficient (b), then use these to break up the middle term.

How to Factor Big Numbers in a Quadratic Equation?

For big numbers, use techniques like the quadratic formula or numerical methods, as manual factoring becomes impractical.

Who Discovered the Quadratic Formula?

Quadratic formula is said to be one of the oldest formulae and its development involved a lot of mathematicians from the ancient world such as Babylonians, Greeks and Indians notably Brahmagupta.